Simplify (4x^3y^-3)/(2x^3x^-1y^2)

Solution:

Step 1: Combine $x$ terms in the denominator

• Identify powers of $x$ to be combined: $\frac{4x^3{y}^{-3}}{2(x^{-1}{x}^3)y^2}$
• Apply the exponent addition rule: $a^m{a}^n=a^{m+n}$
• Simplify the $x$ terms: $\frac{4x^3{y}^{-3}}{2x^{3-1}{y}^2}$

Step 2: Simplify negative exponent for $y$

• Apply the negative exponent rule: $b^{-n}=\frac{1}{b^n}$
• Move $y^{-3}$ to the denominator: $\frac{4x^3}{2x^2{y}^2{y}^3}$

Step 3: Combine $y$ terms in the denominator

• Identify powers of $y$ to be combined: $\frac{4x^3}{2x^2(y^3{y}^2)}$
• Apply the exponent addition rule: $a^m{a}^n=a^{m+n}$
• Simplify the $y$ terms: $\frac{4x^3}{2x^2{y}^{3+2}}$

Step 4: Reduce the fraction by cancelling common factors

• Identify common factors in numerator and denominator: $4$ and $2$
• Factor out the common factor of $2$: $\frac{2(2x^3)}{2(x^2{y}^5)}$
• Cancel out the common factor of $2$: $\frac{2x^3}{x^2{y}^5}$

Step 5: Cancel common $x$ terms

• Identify common $x$ factors: $x^3$ and $x^2$
• Factor out the common $x^2$: $\frac{x^2(2x)}{x^2{y}^5}$
• Cancel out the common $x^2$: $\frac{2x}{y^5}$

Knowledge Notes:

• Exponent addition rule: When multiplying like bases, add the exponents: $a^m{a}^n=a^{m+n}$.

• Negative exponent rule: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent: $b^{-n}=\frac{1}{b^n}$.

• Simplifying fractions: To simplify a fraction, cancel out common factors in the numerator and the denominator.

• Combining like terms: Terms with the same base and exponent can be combined by adding or subtracting the coefficients.

• Reducing fractions: To reduce a fraction to its simplest form, divide both the numerator and the denominator by their greatest common factor (GCF).